To handle the decay of the excited state, we follow a method similar to the Weisskopf-Wigner approximation [1]}:
\({}\begin{aligned}\dot{c}_1(t)&=-\int \frac{d^3 k}{(2\pi )^3}\langle 1_G(t),0|H_I(\sigma _+,a_{\vec{k},x},t)|0_G(t),\gamma _{\vec{k},x}\rangle \exp [-i(\gamma _1(t)+\theta _1(t))+i(\gamma _0(t)+\theta _0(t))]\\&\times \int ^t_0dt^{\prime }\langle 0_G(t^{\prime }),\gamma _{\vec{k},x}|H_I(\sigma _-,a_{\vec{k},x}^{\dagger },t^{\prime })|1_G(t^{\prime }),0\rangle c_1(t^{\prime })\exp [i(\gamma _1(t^{\prime })+\theta _1(t^{\prime }))-i(\gamma _0(t^{\prime })+\theta _0(t^{\prime }))].\end{aligned}\) 
